Alan Turing’s Patterns in Nature, and Beyond

Turing Goes Galactic

Once one starts to look, there seems to be no end to Turing patterns: their forms can be seen in weather systems, the distribution of vegetation across landscapes and even the constellations of galaxies.

Image: Galaxy N51, the Whirlpool Galaxy. (European Space Agency)

Turing Patterns in Cells

Turing patterns can involve not just chemicals, but large, complex systems in which each unit — for example, a cell — is distributed like molecules of pigment.

Pictured is a Turing pattern of cells in Dictyostelium, or a slime mold.

Image: National Institutes of Health

Turing Patterns in 3-D

That markings on animals are produced by Turing systems of pigments is now generally accepted, but the origin of what appear to be Turing patterns in more complex settings — such as limb and tooth and lung development — is still debated.

A basic step towards proving the existence of these three-dimensional Turing patterns is demonstrating a three-dimensional pattern in the lab. In a paper published Feb. 11 in Science, researchers led by Brandeis University chemist Irving Epstein do precisely this.

The researchers set off a chemical reaction that creates Turing patterns in laboratory flasks, then used tomography — a form of imaging used to reconstruct three-dimensional images from thousands of two-dimensional snapshots — to picture them.

"It's an intellectual coup," said Epstein.

Image: Science

How the Leopard Got His Spots

Spots on the coats of jaguars and leopards are thought to be Turing patterns -- but given the near-impossibility of working with these animals in controlled conditions, this hypothesis will probably remain extremely likely but not conclusively tested.

Turing Patterns in a Model Organism

A basic model organism is the zebra fish. By raising them in captivity, tweaking genes or growth conditions, and then seeing what happens, researchers use zebra fish to study how animals develop and function.

Fortunately for researchers studying pattern formation, the markings on zebra fish — which develop from a few juvenile spots, to intricate adult motifs — appeared to fit the Turing model.

In 2008, biologists Akiko Nakamasu and Shigeru Kondo of Japan's Nagoya University used lasers to scar the spots of juveniles in different ways, then watched how their markings developed. These living patterns evolved precisely as computer simulations predicted they would, proving their Turing pattern nature.

Image: In the leftmost two columns are photographs of juvenile and adult zebra fish markings. In the other columns are Turing pattern simulations, developing over time. (Kondo and Nakamasu/Proceedings of the National Academy of Sciences)

Turing Patterns in Fish

At left in each photograph is the eye of a popper fish. At right is a computer-generated image of a pattern generated by a Turing pattern simulation.

Like the seashells, this doesn't prove that the natural markings are Turing patterns of pigments. They could theoretically be produced by some other type of chemical reaction, and just happen to look like Turing patterns.

To be more certain, scientists needed to work with the patterns in living animals — the biological equivalents of chemistry sets.

Proving Their Existence

Even though what appeared to be Turing patterns were immediately evident in nature, it wasn't easy to be sure they were produced by reaction-diffusion systems, rather than some other mechanism.

"It may occur all the time in living systems, but it's hard to definitively show," said Esptein. "You have to identify the activator and the inhibitor, and then to establish certain relationships among the reactions they undergo. In biological systems, it turns out to be quite difficult to unambiguously follow the concentrations."

The breakthrough came during the 1980s, when chemists were able to produce Turing patterns in the laboratory, on thin slabs of gel. In these controlled systems, the reactions could be closely followed, simulated on computers and unambiguously demonstrated as true Turing patterns.

How Turing Patterns Work

At the heart of any Turing pattern is a so-called reaction-diffusion system. It consists of an "activator," a chemical that can make more of itself; an "inhibitor," that slows production of the activator; and a mechanism for diffusing the chemicals.

Many combinations of chemicals can fit this system: What matters isn't their individual identity, but how they interact, with concentrations oscillating between high and low and spreading across an area. These simple units then suffice to produce very complex patterns.

"In principle, the behavior is generic. The trick is that you have to have the right rates for the chemical reactions, the right diffusion rates of reacting species," said Irving Epstein, a Brandeis University chemist who studies pattern formation.

Image: A schematic comparison of the reaction-diffusion model to what's known as the morphogen-gradient model, which is essentially a reaction-diffusion system with two non-interacting chemicals. (S. Miyazama/Science)

Alan Turing's Biology Paper

Near the end of his life, the great mathematician Alan Turing wrote his first and last paper on biology and chemistry, about how a certain type of chemical reaction ought to produce many patterns seen in nature.

First found in chemicals in dishes, then in the stripes and spirals and whorls of animals, so-called Turing patterns abounded. Some think that Turing patterns may actually extend to ecosystems, even to galaxies. That's still speculation — but a proof published Feb. 11 in Science of Turing patterns in a controlled three-dimensional chemical system are even more suggestion of just how complex the patterns can be.

On the following pages, Wired.com takes you on a Turing pattern tour.

Images: Left: Alan Turing. (Ohio State University) Right: Patterns generated by a computer simulation of the Turing model. each is made by the same basic equation, with its parameters slightly tweaked. (Shigeru Kondo & Takashi Miura/Science)